What are the basic trigonometric rule

These define sine, cosine, and tangent based on the sides of a right-angled triangle:

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

These are the reciprocals of the primary ratios:

• Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}$
• Secant: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}$
• Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}$

This relates tangent to sine and cosine:

$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

This is a fundamental identity derived from the Pythagorean theorem:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$